Property (T) discrete quantum groups and subfactors with triangle presentations

TL;DR

This paper establishes a spectral gap criterion for property (T) in discrete quantum groups and subfactors, leading to the construction of new examples that are not derived from classical property (T) groups.

Contribution

It introduces a Zuk-type spectral gap criterion for property (T) and constructs novel property (T) discrete quantum groups via triangle presentations.

Findings

First examples of property (T) discrete quantum groups not related to classical property (T) groups

Develops a spectral gap criterion for property (T) in quantum group setting

Constructs subfactor invariants with property (T) from generators and triangle presentations

Abstract

Kazhdan's property (T) has been studied for several discrete group-like structures, including standard invariants of Jones' subfactors and discrete quantum groups. We prove a Zuk-type spectral gap criterion for property (T) in this setting. This allows us to construct a family of property (T) discrete quantum groups, as well as subfactor standard invariants, given by generators and a triangle presentation. These are the first examples of property (T) discrete quantum groups that are not directly related to a discrete group with property (T).